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A101093 Second partial sums of sixth powers (A001014). 9
1, 66, 860, 5750, 26265, 93436, 278256, 725220, 1703625, 3682030, 7431996, 14167946, 25730705, 44823000, 75305920, 122566056, 193963761, 299373690, 451829500, 668285310, 970507241, 1386109076, 1949746800, 2704487500 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Luciano Ancora, Recurrence relation for the second partial sums of m-th powers

Luciano Ancora, Second partial sums of the m-th powers

C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - N. J. A. Sloane, Mar 23 2014

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

FORMULA

a(n) = n*(1 + n)^2*(2 + n)*(-1 + n*(2 + n))*(-2 + 3*n*(2 + n))/168.

G.f.: x*(1+x)*(1 + 56*x + 246*x^2 + 56*x^3 + x^4)/(1-x)^9. - Colin Barker, Dec 18 2012

a(n) = Sum_{i=1..n} i*(n+1-i)^6, by the definition. - Bruno Berselli, Jan 31 2014

a(n) = 2*a(n-1) - a(n-2) + n^6. - Luciano Ancora, Jan 08 2015

MAPLE

f:=n->(3*n^8-14*n^6+21*n^4-10*n^2)/168;

[seq(f(n), n=0..50)];  # N. J. A. Sloane, Mar 23 2014

MATHEMATICA

CoefficientList[Series[(x+1)(x^4+56x^3+246x^2+56x+1)/(1-x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)

Nest[Accumulate, Range[30]^6, 2] (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 66, 860, 5750, 26265, 93436, 278256, 725220, 1703625}, 30] (* Harvey P. Dale, Jun 05 2019 *)

PROG

(MAGMA) [n*(1+n)^2*(2+n)*(-1+n*(2+n))*(-2+3*n*(2+n))/168: n in [1..40]]; // Vincenzo Librandi, Mar 24 2014

(PARI) vector(30, n, m=n+1; m^2*(3*m^6 -14*m^4 +21*m^2 -10)/168) \\ G. C. Greubel, Aug 28 2019

(Sage) [n^2*(3*n^6 -14*n^4 +21*n^2 -10)/168 for n in (2..30)] # G. C. Greubel, Aug 28 2019

(GAP) List([2..30], n-> n^2*(3*n^6 -14*n^4 +21*n^2 -10)/168); # G. C. Greubel, Aug 28 2019

CROSSREFS

Cf. A000540.

Sequence in context: A223152 A301544 A229328 * A293613 A304838 A056468

Adjacent sequences:  A101090 A101091 A101092 * A101094 A101095 A101096

KEYWORD

nonn,easy

AUTHOR

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

EXTENSIONS

Edited by Ralf Stephan, Dec 16 2004

STATUS

approved

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Last modified October 13 20:38 EDT 2019. Contains 327981 sequences. (Running on oeis4.)